String calculation of the long range Q Q potential

Transcription

1 String calculation of the long range Q Q potential Héctor Martínez in collaboration with N. Brambilla, M. Groher (ETH) and A. Vairo April 4, 13

2 Outline 1 Motivation The String Hypothesis 3 O(1/m ) corrections 4 The spectrum 5 Outlook

3 Motivation

4 The static quark-antiquark potential is given by where is the rectangular Wilson loop V () (r) = i lim T,T W T ln W { } W P exp ig dz µ A µ(z), r T W

5 In the long distance limit this leads to a static potential with a linear dependence in r with force lines V () = σr

6 The string hypothesis or Effective String Theory (EST) states that at long distances (rλ QCD >> 1) lim W (T T W W, r) = Z Dξ 1 Dξ e is string(ξ 1,ξ ) where S string = dtdz L( µ ξ l ) = σ dtdz (1 1 µξl µ ξ l ), and σ is the string tension. Using these relations Luscher et al. (Nucl. Phys. B173, 365, 198) obtained a non trivial correction to the static potential V () (r) = σr + µ π 1r This result agrees with lattice data for σ.1 GeV (Luscher et al. JHEP, 7, )

7 From the matching between NRQCD and pnrqcd Brambilla Pineda Soto and Vairo (PRD63, 143, ) obtained the relativistic corrections to the static potential at the order of 1/m: given by V = V () + V (1,) ( 1 m m ) +... V (1,) (r) = 1 lim T where W / W and T dt t ge 1(t) ge 1() c, O 1(t 1)O (t ) c = O 1(t 1)O (t ) O 1(t 1) O (t )

8 Using symmetry considerations the following mapping between the EST and the operator insertions can be applied (Perez-Nadal and Soto, PRD79, 114, 9 & Kogut and Parisi PRL47, 189, 1981) ψ (t)e l (t, r )ψ(t) Λ zξ l (t, r ) χ (t)e l (t, r )χ(t) Λ zξ l (t, r ) ψ (t)b l (t, r )ψ(t) Λ ɛ lm t zξ m (t, r ) χ (t)b l (t, r )χ(t) Λ ɛ lm t zξ l (t, r ) ψ (t)e 3 (t, r )ψ(t) Λ χ (t)e 3 (t, r )χ(t) Λ ψ (t)b 3 (t, r )ψ(t) Λ ɛ lm t zξ l (t, r ) zξm (t, r ) ψ (t)b 3 (t, r )ψ(t) Λ ɛ lm t zξ l (t, r ) zξm (t, r ).

9 Using this mapping for the 1/m correction to the potential we get E l (t, r )Em (, r ) c Λ4 z z ξ l (t, r/)ξ m (, r/) = Λ 4 z z G lm F (t, r/;, r/) where the mapping into the EST is expressed in terms of the two point correlator of the string degrees of freedom: ( G lm F (it, z; it, z ) = δlm cosh( π 4πσ ln (t r t )) + cos( π (z + ) r z )) cosh( π (t r t )) cos( π (z r z )) then the potential is given by V (1,) (r) g Λ 4 = g Λ 4 σπ lim ɛ + ɛ lim ɛ + πɛ r dt dt t z z G ll F(t, r/;, r/) t sinh (t) V (1,) = g Λ 4 σπ ln( σr) + Infinite constant

10 (Perez-Nadal and Soto, PRD79, 114, 9)

11 O(1/m ) corrections to the potential

12 The O(1/m ) corrections are organized as follows where ( V = V () + V (1,) ) + V(,) + V(,) + V(1,1) m 1 m m 1 m m 1m and = SD + SI, V (,) = V (,) SD + V (,) SI, SI = 1 { } p 1, (r) + V(,) (r) L L p r 1 + V r (,) (r), p 1 = i x1 and L 1 = r p 1

13 For the spin-dependent part we have SD = LS (r) L 1 S, where S is the total spin operator. The same kind of decomposition is made for the potential where = SD + SI, SI = 1 { } p 1 p, (r) V(1,1) (r) L (L p r 1 L + L L 1 ) + V r (1,1) (r) SD = L 1 S (r)l 1 S L S 1 (r)l S 1 + S 1 S (r)s 1 S + (r)s 1 r, S 1 and S 1 ( r) = 3 r σ 1 r σ σ 1 σ L 1 S (r) = L S 1 (r, m 1 m ).

14 Let us consider for instance where LS (r) = c(1) F ir lim r The mapping leads to and therefore SD = LS (r) L 1 S, T T dt gb 1(t) ge 1() + c(1) S r r ( rv() ), B(t, r ) E(, r ) 3 Λ Λ t z z G F(t, r/;, r/) LS (r) = i g c (1) F Λ Λ lim σr = g c (1) F Λ Λ kr ɛ + ɛπ r + 4c(1) F g Λ Λ ɛ UVπr dt t cos(t) sin 3 (t) + c(1) S σ r + c(1) S σ r

15 Renormalized the potential reads Poincaré invariance implies LS (r) = g c (1) F Λ Λ µc σr r 1 dv () + LS L r dr S 1 =, Gromes, Z. Phys. C 6, 41 (1984) Which translates in to the parameters as r dv () + =. dr L L Brambilla et al. Nuovo Cimiento A 13, 59 (199) µ c = σ/ gλ = σ So the unknown constant we have introduce in order to renormalize LS (r) is now fixed.

16 r (r) = πc f α sc (1) D δ (3) (x 1 x ) ic(1) T F lim dt gb 1 (t) gb 1 () c T ( r V(,) p ) i T t1 t lim dt 1 dt dt 3 (t t 3 ) ge 1 (t 1 ) ge 1 (t )ge 1 (t 3 ) ge 1 () c T + 1 ( T t1 ) i r lim dt 1 dt (t 1 t ) ge i T 1 (t 1)gE 1 (t ) ge 1 () c i ( T t1 i r V()) lim dt 1 dt (t 1 t ) 3 ge i T 1 (t 1)gE 1 (t ) ge 1 () c + 1 ( T ) i r lim dt t 3 ge i 4 T 1 (t)gej 1 () c( j r V() ) i T lim dt t 4 ge i 1 T 1 (t)gej 1 () c( i r V() )( j r V() ) d (1) 3 f abc d 3 x lim T W g Ga µν (x)gb µα (x)gc να (x),

17 The potentials in the EST read V () (r) = σr + µ π 1r, V (1,) (r) = g Λ 4 σπ ln( σr) p (r) =, p (r) =, L (r) = g Λ 4 r 6σ, L (r) = g Λ 4 r 6σ LS (r) = g c (1) F Λ Λ σ r µc r L S (r) = g c (1) F Λ Λ 1 σ r, L 1 S (r) = g c (1) F g Λ Λ σ r, S (r) = g (1) cf c() F Λ π 3 45σ 4 r 5 S 1 (r) = 1 4 V(1,1) S r (r) =.44 g4 Λ 8 r σ π 3 r (r) =.61 g4 Λ 8 r σ π 3 g (1) c F Λ σπɛ 3 + 3g Λ 4 UV 9g Λ 4 ζ(3) 4π 3 σ π 3 σ ζ(3)( i r r i r V() ) g Λ 4 r 3 i r (r i r V() ) + 7g Λ 4 r 3 18σ ( i r V() )( i r V() ) 45σ ( i r V() )( i r V() )

18 The Spectrum

19 We will only consider the leading terms up to 1/r in the equal mass case V () (r) = σr V (1,) (r) = σ π ln( σr) p (r) =, p (r) =, L (r) = σr 6, L (r) = σr 6 LS (r) = σ r L S 1 (r) = L 1 S (r) = S (r) = S 1 (r) = r (r) = σ r( σr ) r (r) = σ r( σr )

20 Then the long range potential simplifies to V(r) = σr + m V(1,) + 1 L m r L + LS L S + r + r = σr + 1 { } σ m π ln( σr) + 1 { σ6r m L σr } L S.58 σ r.11 σ 3 r 3

21 String Spectrum for m=1, 11, Energy S (1 3 S 1) 1 S ( 3 S 1) 3 PJ 3 3 DJ

22 Outlook: Lattice QCD Yoshiaki Koma and Miho Koma: Heavy quarkonium spectroscopy in pnrqcd with lattice QCD input arxiv: Phenomenology of quarkonium radiative transitions Brambilla, Pietrulewicz, Vairo: Model-independent Study of Electric Dipole Transitions in Quarkonium Phys. Rev. D 85, 945 (1)

23 Thanks for your attention!

24 Backup

25 V (1,) (r) = 1 T lim dt t ge 1 (t) ge 1 () c, T i T p (r) = ˆriˆr j lim dt t ge i T 1 (t)gej 1 () c, i ( L (r) = δ ij j) T 3ˆr iˆr lim dt t ge i 4 T 1 (t)gej 1 () c, LS (r) = c(1) T F ir lim dt t gb r 1 (t) ge 1 () + c(1) S T r r ( rv() ), i T p (r) = ˆriˆr j lim dt t ge i T 1 (t)gej 1 () c, i ( L (r) = δ ij j) T 3ˆr iˆr lim dt t ge i 4 T 1 (t)gej 1 () c, L S (r) = c(1) T F ir lim dt t gb 1 r 1 (t) ge () T S (r) = c (1) F c() T F i lim dt gb 1 (t) gb () 4(d sv + d vvc f ) δ (3) (x 1 x ), 3 T (r) = c(1) F c() F iˆr iˆr [ ] T j lim dt gb i S 1 4 T 1 (t)gbj δij () 3 gb 1(t) gb (),

26 and... r (r) = πc f α sc (1) D δ (3) (x 1 x ) ic(1) T F lim dt gb 1 (t) gb 1 () c T ( r V(,) p ) i T t1 t lim dt 1 dt dt 3 (t t 3 ) ge 1 (t 1 ) ge 1 (t )ge 1 (t 3 ) ge 1 () c T + 1 ( T t1 ) i r lim dt 1 dt (t 1 t ) ge i T 1 (t 1)gE 1 (t ) ge 1 () c i ( T t1 i r V()) lim dt 1 dt (t 1 t ) 3 ge i T 1 (t 1)gE 1 (t ) ge 1 () c + 1 ( T ) i r lim dt t 3 ge i 4 T 1 (t)gej 1 () c( j r V() ) i T lim dt t 4 ge i 1 T 1 (t)gej 1 () c( i r V() )( j r V() ) d (1) 3 f abc d 3 x lim T W g Ga µν (x)gb µα (x)gc να (x),

27 r (r) = 1 ( r V(1,1) p ) T t1 t i lim dt 1 dt dt 3 (t t 3 ) ge 1 (t 1 ) ge 1 (t )ge (t 3 ) ge () c T + 1 ( T t1 ) i r lim dt 1 dt (t 1 t ) ge i T 1 (t 1)gE (t ) ge () c + 1 ( T t1 ) i r lim dt 1 dt (t 1 t ) ge i T (t 1)gE 1 (t ) ge 1 () c i ( T t1 i r V()) lim dt 1 dt (t 1 t ) 3 ge i T 1 (t 1)gE (t ) ge () c i ( T t1 i r V()) lim dt 1 dt (t 1 t ) 3 ge i T (t 1)gE 1 (t ) ge 1 () c + 1 ( T i r lim dt t 3 { ge i } ) 4 T 1 (t)gej () c + gei (t)gej 1 () c ( j r V() ) i T lim dt t 4 ge i 6 T 1 (t)gej () c( i r V() )( j r V() ) + (d ss + d vsc f ) δ (3) (x 1 x ), r (r) =.44 σ r π 3 r (r) =.61σ r π 3 + 6σ r 3 σ r 3 π 3 ζ(3) 45 9σ r 3 7σ r 3 π 3 ζ(3) + 9